Measuring Market Risk Second Edition Kevin Dowd Measuring Market Risk For other titles in the Wiley Finance Series please see www.wiley.com/finance Measuring Market Risk Second Edition Kevin Dowd Copyright C 2005 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wiley.com All Rights Reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, 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D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Library of Congress Cataloging-in-Publication Data Dowd, Kevin Measuring market risk / Kevin Dowd.—2nd ed p cm Includes bibliographical references and index ISBN 13 978-0-470-01303-8 (cloth : alk paper) ISBN 10 0-470-01303-6 (cloth : alk paper) Financial futures—Mathematical models Risk management—Mathematical models Portfolio management—Mathematical models I Title HG6024.3.D683 2005 332.63 2042—dc22 2005010796 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 13 978-0-470-01303-8 (HB) ISBN 10 0-470-01303-6 (HB) Typeset in 10/12pt Times by TechBooks, New Delhi, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production Contents Preface to the Second Edition xiii Acknowledgements xix The Rise of Value at Risk 1.1 The emergence of financial risk management 1.2 Market risk measurement 1.3 Risk measurement before VaR 1.3.1 Gap analysis 1.3.2 Duration analysis 1.3.3 Scenario analysis 1.3.4 Portfolio theory 1.3.5 Derivatives risk measures 1.4 Value at risk 1.4.1 The origin and development of VaR 1.4.2 Attractions of VaR 1.4.3 Criticisms of VaR 5 9 11 13 Appendix: Types of Market Risk Measures of Financial Risk 2.1 The mean–variance framework for measuring financial risk 2.2 Value at risk 2.2.1 Basics of VaR 2.2.2 Determination of the VaR parameters 2.2.3 Limitations of VaR as a risk measure 2.3 Coherent risk measures 2.3.1 The coherence axioms and their implications 2.3.2 The expected shortfall 2.3.3 Spectral risk measures 15 19 20 27 27 29 31 32 32 35 37 vi Contents 2.3.4 Scenarios as coherent risk measures 2.4 Conclusions 42 44 Appendix 1: Probability Functions 45 Appendix 2: Regulatory Uses of VaR 52 Estimating Market Risk Measures: An Introduction and Overview 3.1 Data 3.1.1 Profit/loss data 3.1.2 Loss/profit data 3.1.3 Arithmetic return data 3.1.4 Geometric return data 3.2 Estimating historical simulation VaR 3.3 Estimating parametric VaR 3.3.1 Estimating VaR with normally distributed profits/losses 3.3.2 Estimating VaR with normally distributed arithmetic returns 3.3.3 Estimating lognormal VaR 3.4 Estimating coherent risk measures 3.4.1 Estimating expected shortfall 3.4.2 Estimating coherent risk measures 3.5 Estimating the standard errors of risk measure estimators 3.5.1 Standard errors of quantile estimators 3.5.2 Standard errors in estimators of coherent risk measures 3.6 The core issues: an overview 53 53 53 54 54 54 56 57 57 59 61 64 64 64 69 69 72 73 Appendix 1: Preliminary Data Analysis 75 Appendix 2: Numerical Integration Methods 80 Non-parametric Approaches 4.1 Compiling historical simulation data 4.2 Estimation of historical simulation VaR and ES 4.2.1 Basic historical simulation 4.2.2 Bootstrapped historical simulation 4.2.3 Historical simulation using non-parametric density estimation 4.2.4 Estimating curves and surfaces for VAR and ES 4.3 Estimating confidence intervals for historical simulation VaR and ES 4.3.1 An order-statistics approach to the estimation of confidence intervals for HS VaR and ES 4.3.2 A bootstrap approach to the estimation of confidence intervals for HS VaR and ES 4.4 Weighted historical simulation 4.4.1 Age-weighted historical simulation 4.4.2 Volatility-weighted historical simulation 4.4.3 Correlation-weighted historical simulation 4.4.4 Filtered historical simulation 83 84 84 84 85 86 88 89 89 90 92 93 94 95 96 Contents vii 4.5 Advantages and disadvantages of non-parametric methods 4.5.1 Advantages 4.5.2 Disadvantages 4.6 Conclusions 99 99 100 101 Appendix 1: Estimating Risk Measures with Order Statistics 102 Appendix 2: The Bootstrap 105 Appendix 3: Non-parametric Density Estimation 111 Appendix 4: Principal Components Analysis and Factor Analysis 118 Forecasting Volatilities, Covariances and Correlations 5.1 Forecasting volatilities 5.1.1 Defining volatility 5.1.2 Historical volatility forecasts 5.1.3 Exponentially weighted moving average volatility 5.1.4 GARCH models 5.1.5 Implied volatilities 5.2 Forecasting covariances and correlations 5.2.1 Defining covariances and correlations 5.2.2 Historical covariances and correlations 5.2.3 Exponentially weighted moving average covariances 5.2.4 GARCH covariances 5.2.5 Implied covariances and correlations 5.2.6 Some pitfalls with correlation estimation 5.3 Forecasting covariance matrices 5.3.1 Positive definiteness and positive semi-definiteness 5.3.2 Historical variance–covariance estimation 5.3.3 Multivariate EWMA 5.3.4 Multivariate GARCH 5.3.5 Computational problems with covariance and correlation matrices Appendix: Modelling Dependence: Correlations and Copulas Parametric Approaches (I) 6.1 Conditional vs unconditional distributions 6.2 Normal VaR and ES 6.3 The t-distribution 6.4 The lognormal distribution 6.5 Miscellaneous parametric approaches 6.5.1 L´evy approaches 6.5.2 Elliptical and hyperbolic approaches 6.5.3 Normal mixture approaches 6.5.4 Jump diffusion 127 127 127 128 129 131 136 137 137 138 140 140 141 141 142 142 142 142 142 143 145 151 152 154 159 161 165 165 167 167 168 viii Contents 6.6 6.7 6.8 6.9 6.5.5 Stochastic volatility approaches 6.5.6 The Cornish–Fisher approximation The multivariate normal variance–covariance approach Non-normal variance–covariance approaches 6.7.1 Multivariate t-distributions 6.7.2 Multivariate elliptical distributions 6.7.3 The Hull–White transformation-into-normality approach Handling multivariate return distributions with copulas 6.8.1 Motivation 6.8.2 Estimating VaR with copulas Conclusions Appendix: Forecasting Longer-term Risk Measures 169 171 173 176 176 177 177 178 178 179 182 184 Parametric Approaches (II): Extreme Value 7.1 Generalised extreme-value theory 7.1.1 Theory 7.1.2 A short-cut EV method 7.1.3 Estimation of EV parameters 7.2 The peaks-over-threshold approach: the generalised Pareto distribution 7.2.1 Theory 7.2.2 Estimation 7.2.3 GEV vs POT 7.3 Refinements to EV approaches 7.3.1 Conditional EV 7.3.2 Dealing with dependent (or non-iid) data 7.3.3 Multivariate EVT 7.4 Conclusions 189 190 190 194 195 201 201 203 204 204 204 205 206 206 Monte Carlo Simulation Methods 8.1 Uses of Monte carlo simulation 8.2 Monte Carlo simulation with a single risk factor 8.3 Monte Carlo simulation with multiple risk factors 8.4 Variance-reduction methods 8.4.1 Antithetic variables 8.4.2 Control variates 8.4.3 Importance sampling 8.4.4 Stratified sampling 8.4.5 Moment matching 8.5 Advantages and disadvantages of Monte Carlo simulation 8.5.1 Advantages 8.5.2 Disadvantages 8.6 Conclusions 209 210 213 215 217 218 218 219 220 223 225 225 225 225 Applications of Stochastic Risk Measurement Methods 9.1 Selecting stochastic processes 9.2 Dealing with multivariate stochastic processes 227 227 230 376 Bibliography Schachter, B (1998) ‘The value of stress testing in market risk management.’ Pp 5F-0–5F-11 in T Haight (ed.) 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The case of 15 time series from emerging markets.’ RiskMetricsTM Monitor, Fourth Quarter: 20–32 Zangari, P (1997) ‘What risk managers should know about mean reversion and jumps in prices.’ RiskMetricsTM Monitor, Fourth Quarter: 12–41 Zangari, P (1998) ‘Exploratory stress-scenario analysis with applications to EMU.’ RiskMetricsTM Monitor Special Edition: 30–53 Index Abken, P A., 233 Acerbi, C, 33–35, 37, 39, 72, Albanese, C., 258 Adaptive kernel, 86, 115, 117 Adaptive meshes, 256 Additivity, 271–273 Adelman, I., 123 AGARCH, 97, 131, 135 Age-weighted HS, 93–95, 99 Alexander, C., 132, 143–144, 285 ALM (asset-liability management), 241, 362 Algorithmic methods (for options VaR), 249, 251, 253, 264 Allied Irish Bank, Almgren, R., 313 Alternative hypothesis, 323, 325–327, 334–336, 341, 344 American options, 215, 253–256 Analytical methods (for options VaR), 249–253, 264 Andersen, L., 225 Andersen, T G., 137, 187 Anderson, G., 345 Anderson-Darling test, 344, 348–349 Andrews, D K., 109–110 Annuities, annuity risks, etc., 242–248 Antithetics, 218, 223, 255 APT (Arbitrage Pricing Model), AR(1) process, 336 Aragon´es, J R., 265 Archimedean copula, 148 ARIMA, 134 Arithmetic Ornstein-Uhlenbeck process with jumps, 229 Arithmetic returns, 54–56, 59–61, 63, 154, 184 ARMA, 134, 137 ART (Alternative Risk Transfer), 225, 241 Artzner, P., 32, 34–35 Ashton, D., 170 Ask price, 310 Asymptotic power law, 49 Autocorrelation, 98, 330 ‘Average tail VaR’ (or ‘average quantile’, etc.) approach, 64, 66, 68, 83, 116, 212 Backtesting, backtesting, etc., 29, 52, 311, 321–343, 347, 352, 359, 363 Backtesting backtests, 341–342 Backtesting chart, 322–323, 342 Bandwidth, 114–115 Bangia, A., 312 Bank of Tokyo-Mitsubishi, 355 Barings Bank, 4, 15, 356, 361 Barone-Adesi, G., 96–98 Barraquand, J., 223 Bartlett test, 343 Bartlmae, K., 99 Basak, S., 14 Basel Accord, 52, 184, 291, 328 Basel II, 52 Basel capital adequacy rules, 30, 354, 328 Bassett, G., 172 Bassi, F., 197 Bauer, C., 167, 177 Baumol, W J., 27 Bauwens, L., 143 BDS test, 330 Beder, T., 13, 304–305, 353 ‘Before and after’ approach (to IVaR), 266–268, 271, 274 Beirlant, J., 189 BEKK model, 143 Benchmark backtesting, 340 Berkowitz, J., 297, 314, 333–334, 353 Berkowitz transformation, etc., 333–335, 342 Bertsimas, D., 313 ‘Best hedge’, 266, 276 Best replicating portfolios, 276–277 380 Index Beta, 7, 8, 270, 282–283 Beta distribution, 243 Bias, 108, 110, 114, 137, 200, 207, 223 Bickel, P J., 170 Bid-ask spread, 294, 310–313, 319 Bid price, 310 Billio, M., 169 Binary regression, 330 ‘binocdf’ function (in MATLAB), 325, 327 ‘binofit’ function (in MATLAB), 326, 328 Binomial distribution, 241, 325–326 Binomial test, 324–326, 342 Bins, binned data, etc., 71–72, 86, 111–113, 344–345 BIS, 2, 3, 296, 319 Bisection method, 137, 180–182 Black-Derman-Toy model, 355 Black-Scholes, 137, 209–212, 252–253, 256, 258, 302–303, 351, 354 Blake, D., 244, 246 Blanco, C., 229, 338 Block approach (bootstrap), 110 Block maxima approach, 204–205 Bollen, B., 137 Bollerslev, T., 131, 137 ‘bondvares’ function (in MMR Toolbox), 238 Bootstrap, 73, 83, 85–86, 90–91, 95, 97–98, 105–110, 116, 164–165, 200, 298–299, 340, 342 ‘bootstrapesfigure’ (in MMR Toolbox), 91 ‘bootstrapvarfigure’ (in MMR Toolbox), 90 Bootstrapped HS, 85–86, 90–91 Bootstrapped VaR, 90 Bootstrapped ES, 90–91 Borse, G J., 81 Bouchaud, J P., 260–261 Bouchaud-Potters dominant factor approach, 260–261 Boudoukh, J., 93 Bouy´e, E., 149 Box-Cox transformation, 170 Box kernel, 86, 114, 116 Box-Pierce test, 132, 134, 330 Boyle, P., 211, 218–219, 224, 274 Brace-Gatarak-Musiela model, Breckling, J., 167 Breuer, T., 297 Britten-Jones, M., 260, 262–263, 304 Britten-Jones/Schaefer delta gamma approach, 260 Broadie, M., 215, 218–219, 222–223 Brock, W A., 330 Brown-Forsythe test, 343 Brownian bridge methods, 222, 255 Brusilovskiy, P., 340 BRW (age-weighting) approach, 93–95, 99 Buchinsky, M., 109–110 Butler, J S., 87, 90, 116 Byström, H N E., 143 BZW 355 Cairns, A J G., 351 Cap Gemini, 361 Capital allocation, requirements, etc., 9, 12, 29, 34, 52, 265, 241, 273–274, 291, 293, 306 CAPM, 8, 12, 23, 135 Cárdenas, J., 219, 258 Cashflow risk, 11; see also under Liquidity risk Cash flow at risk (CFaR), 316; see also under Liquidity at risk Cauchy distribution, 50, 160, 165 CAViaR, 172–173 Central limit theorem, 21, 51, 154, 165, 183, 190, 206–207, 221, 239 Central tendency, 46–47, 89, 190 Cerny, A., 43 Chappell, D., 164 Characteristic function, 47–48 Chebyshev inequality, 263–264 Chernozhukov, V., 172 Cherubini, U., 147, 171, 316 Chibumba, A., 143 Chicago Board Options Exchange, Chicago Mercantile Exchange, 3, 42, 291 Chief Risk Officer (CRO), 362–363 Chien, C.-C C., 170 Chi-squared distribution, 160, 260, 263, 330–331, 358–359 Chi-squared tests, 106, 304, 330–331, 333–335, 343–345 Choleski decomposition, matrix, 95–96, 216–217, 304 Chriss, N A., 313 Christoffersen, P F., 329–330, 336 Christoffersen backtest, 329–330, 342 Cleaning data, 118, 321–323, 354 Clewlow, L., 211, 218, 230 Coherent risk measure, 14, 19–20, 32, 37, 39, 42–44, 53, 64, 66–68, 72–73, 80, 86, 152, 265, 277, 291, 297, 307 Coherency axioms, 33, 43 Collinearity problems, 280, 282 Commodity positions, 12, 280 Comonotonic, comonotonicity, etc., 39, 146 Component ES (CES), 277 Component risks, 265–277 Component GARCH, 135 Conditional distribution, conditionality, etc., 25–26, 45–46, 74, 111, 128, 132, 151–153, 157–158, 204–205, 224, 228 Conditional (dynamic) EV, 204–205 Index Conditional Monte Carlo simulation, 224 Conditional VaR, 35 Confidence interval, 69–73, 83, 89–91, 98, 102–106, 108, 110, 116, 152, 164–165, 207, 212, 221, 225, 244, 255, 261, 306, 325, 340–341344, 346, 357–358 Confidence region, 262–263, 304 Constant spread approach, 311–312 Cont, R., 167 Context modelling, 99 Contingency planning, 295, 306, 363 Control variates, 218–219, 255–256 Convertible bonds, 356 Convexity, 6, 12, 37, 257, 259, 298, 300 Copulas, 51, 74, 145–150, 151, 178–183, 206, 230 Cornish-Fisher approximation, 171–172, 271 Cornish-Fisher VaR, 171–172 Correlation, 7, 10, 51, 95, 96, 100, 106–107, 118, 122, 125, 127, 135, 137–139, 140–142, 145, 148, 168, 174–178, 181, 206, 209, 216, 218, 225, 230, 231, 269, 271–272, 275, 279, 281–285, 292, 295–296, 302, 351, 353, 356, 358 Correlation-weighted HS, 95–96, 99 Corrigan, E G., Corsi, F., 137 Cosandey, D., 314 Cotter, J., 189 Countermonotonicity, 146 Counterparty risk, 319 Coupon bonds, 287–289 Covariance, 95, 119–120, 122, 133, 137–138, 140–145, 151, 206, 282–283, 287, 319, 358 Covariance matrix, 49, 74, 83, 95–96, 98–99, 118–121, 123–124, 127, 142–144, 151, 173–174, 176–178, 268, 279, 281, 283, 297, 301, 354, 356 Cox-Ingersoll-Ross model, 228, 237–238 Cram´er-von Mises test, 344 CrashMetrics, 305–306, 319 Credit derivatives, 15, 225, 239, 317 Credit risk(s), credit-related risks, etc., 1, 11, 13, 15, 52, 227, 238–240, 280, 291–293, 296, 299, 317–318, 320 Crisis-related liquidity risks, 309, 319–320 Crnkovic, J., 348 Crouhy, M., 239, 328, 361 Cumulative density function (or cdf), 22, 45, 56, 69, 102, 116 Cumulative histogram(or empirical cdf), 56, 58, 69–70 ‘Curse of dimensionality’, 83, 99, 206, 256 CVaR (component VaR), 265, 271–276 Cvitanic, J., 43 381 D’ Vari, R., 237 Daiwa, Danielsson, J., 14, 189, 200 Dav´e, R D., 144 Davison, A C., 107–108 DB (defined benefit) pension scheme, 244–247 ‘dbpensionvar’ function (in MMR Toolbox), 246 DC (defined contribution) pension scheme, 244, 246–248 ‘dcpensionvar’ function (in MMR Toolbox), 247–248 de Vries, C G., 189, 200 Default risk, 15, 238–240, 284, 317 ‘defaultriskybondvar’ function (in MMR Toolbox), 240 della Lunga, G., 171, 316 Delta, 8, 211, 218–219, 257–258, 274, 294, 300, 306, 355 Delta-gamma approximations, 218, 249, 256–264, 290, 298, 303, 305, 319 Delta-gamma methods for options VaR, 218, 220, 249, 256–264 Delta normal methods for options VaR, 257–258, 260 ‘delVaR’ (approach to IVaR), 267–270, 274 Dembo, R., 12 Denecker, K., 99 Dependence, 51, 110, 145–150, 177–178, 180, 183, 204–206, 230, 336 Derman, E., 351–352, 354, 356 Derivatives Policy Group, 4, 298 Diagonal matrix, 119 Dias, M A G., 230 Diebold, F X., 195, 207 Diff option, 141 Diff swaps, 125 Dimensionality, dimensionality problems, etc., 83, 118, 121–123, 143, 206, 209, 221–225, 227, 230, 248, 279, 280, 356 Dirac delta function, 38 Discretisation errors, 228, 237, 356 Dispersion (measures), 26, 46–47 Distortion risk measures, 43–44 Distribution-equality tests, 331–336, 343–344 Diversification effects, 175–176, 275–276, 283, 354–355 Docksum, K A., 170 Domains of attraction, 51, 165, 191, 201 Dominant factors, 260 Double-bootstrap, 109 Dowd, K., 12, 164, 179, 187, 265, 270, 285, 334, 357 Downside semi-variance, 26 Drachman, J., 348 ‘Drill-down’ capability, 274–275 Drost F., 186 382 Index Drzik, J., 16 Duffie, D., 95, 168–169 Duration, 5–6, 11, 257, 298, 300 Duration analysis, 5–6 Dynamic hedging, 8–9, 14, 219, 291, 353 Dynamic risk measures, 43 Dynamic risks, 227, 234–236 Early exercise, 225–227, 253–256, 264, Eberlein, E., 167, 169, 177 Efficient frontier, 22–23 Efron, B., 105, 107–108 EGARCH, 131, 135 Eigenvalue, 118–119, 121,124, 143–144, 232 Eigenvalue decomposition, 216 Eigenvector, 118–119, 231 Elliptical copulas, 148 Elliptical distributions, 19–20, 22, 24, 26, 32, 34, 44, 49–51, 138, 146, 150, 154, 167, 206 Elton, E J., 20 Embrechts, P., 79, 149, 189–190, 196, 201, 203, 205 Empirical distribution, 64, 78, 343 Endogenous liquidity, 310, 315 Endogenous price approaches, 314 Endogenous model risk, 356, 361 Energy prices, risks, etc., 227–231 Engle, R F., 131, 135, 143, 172, 330 Enron, 4, 299 Enterprise-wide risk management (ERM), 20, 179 Entropy measure, 26 Epanechinikov kernel, 86, 114, 116–117 Equal-weighted volatility, See under Historical volatility Equity indices, 281–283 Equity positions, etc., 15, 280, 282–284, 290, 300, 340 Equity VaR, 282–283 ES (Expected shortfall), 26, 35–39, 42, 52, 54, 64–67, 69, 72–74, 83–86, 88–93, 95, 98–100, 104, 116, 119, 121, 124, 151–152, 154–157, 165, 167, 173–174, 176–177, 190, 194, 202, 204, 220, 238, 241–242, 246, 255, 277, 292, 293–295, 297, 304, 319–320, 338 ES-based decision rules, 36 ES distribution function, 89 Estrella, A., 258 Euler method, 214–215 Euler’s theorem, 272–273, 277 Evans, M., 160 EV (Extreme-value) copulas, 148 EV (Extreme-value) distribution, 50–51, 192, 194 EV (Extreme-value) methods, 152, 159, 207, 319, 339 EVT (Extreme-value theory), 51, 148, 152, 157, 160, 189–190, 197, 201, 204–207, EWMA (Exponentially weighted moving average), 94, 127, 129–132, 135, 137, 140, 142, 152, 356 ‘example10point4’ (in MMR Toolbox), 256 Excel, 21, 56, 84, 284, 287 Excess kurtosis, 25, 49–50, 75, 132, 151, 153, 157, 159–160, 168–169, 177, 214, 312, 334 Exogenous liquidity, 310, 315 Exogenous spread approach, 312–313 Expected tail loss, 35 Expected utility theory, etc., 32, 36–37, 315 Exponential spectral risk measure, 40–41, 66–67 Exponential risk-aversion (or weighting) function, 40–41, 66–67 Extremal index, 205 Extreme VaR, 190, 192–193 Extremes, 104, 157, 189–193, 206–207, 353 F-tests, 343 Fackler, P L., 81, 179 Factor analysis, 118, 122–125 Factor GARCH, 135 Factor loadings, 123 Factor push analysis, 292, 303–305 Faddy, M J., 170 Fallon, W., 258 Fama, E F., 165 Fast Fourier transform methods, 148, 167 Federal Reserve Bank of New York, Feuerverger, A., 258 Figlewski, S., 128, 256 Filter-rule strategy, 234–236 Filtered Historical Simulation (FHS), 96–99, 106 Financial engineering theory, 287, 290 Fishburn, P C., 39 Fishburn measure, 26 Fischer, T., 277 Fisher-Tippett theorem, 148, 190 Fixed-income positions, etc., 5, 12, 15, 118, 257, 227, 230, 236–238, 241, 280–281, 340 Fixed-income VaR, 282 ‘Flight to quality’, 300 Fong, H G., 249, 251 Foreign-exchange positions etc., 280–282, 300, 304, 353 Forint, 282 Forward rate agreements (FRAs), 287 Forward value, 53 Forward contracts, 249, 286, 290 ‘4:15 Report’, 9–10 Fourier transform, 48 Fr´echet distribution, 191, 193–195, 201 Fr´echet VaR, 193–194 Frequency (of exceedance) tests, 324–328, 342–344 Frey, R., 205 Index Froot, K., 309 Futures contracts, 3, 15, 236, 249, 280, 286, 290, 317–318 Fuzzy logic, 171 G-30 Report, 4, 10 GAAP, 16 ‘Gaming’ VaR models, 32, 361 Gamma, 8, 249, 258–260, 262, 300, 306 Gamma function, 170, 196 Gao, B., 256 Gap analysis, 4, 318 GARCH, 83, 94, 95, 97–98, 110, 127–129, 131–135, 137, 140, 142–143, 152–153, 156, 163, 166, 169, 177, 186–187, 204–205, 238, 353–354 Garman, M B., 265, 267, 270 Gauge theory, 170 Gauss-Legendre methods, 81 Gaussian copula, 181 Gaussian kernel, 86, 114–116 General Accounting Office (US), General insurance risks, 241–242 Generalised error distribution, 170 Generalised hyperbolic distributions, 50, 167 Generalised lambda distribution, 170 Generalised Pareto distribution, 64, 201 ‘Generalised scenarios’, 42–43 Genetic algorithm, 173 Gentle, J E., 123 GEV (Generalised Extreme) distribution, 190–192, 201, 205 GEV (Generalised Extreme Value) theorem, 190, 201 GEV (Generalsed Extreme-Value) theory, 190–191, 201, 202–204 Geometric Brownian motion (GBM), 50, 136–137, 161–163, 210, 213–217, 225, 228, 230–231, 315–316, 353 Geometric distribution, 324 Geometric returns, 54–56, 61–63, 154, 161–163, 184 Ghost effect, 93–95, 100, 128–130, 140, 142 Giacomini, E., 336 Giannopoulos, K., 96–98, 132 Gibbs sampler, 168, 224 Gibson, M S., 168 Gini coefficient, 26 Giot, P., 170, 311, 316 Glasserman, P., 177, 215, 218–223 Glosten, L., 135 Gnedenko-Pickands-Balkema-deHaan (GPBdH) theorem, 202–203 Gottschling, A., 99 Grammig, J., 311, 316 Granger, C W J., 173 383 Greek parameters, 8, 12, 52, 209–211, 218–219, 256, 291, 305–306, 319–320 Greenspan, A., 294 Grootveld, H., 32, 39 Gruber, M J., 20 Guilder, 282 Guldimann, T., 10 Gumbel, E J., 196 Gumbel copula, 148–149 Gumbel distribution, 24, 47, 103, 191–195 Gumbel VaR, 192 Hallerbach, W G., 32, 39, 265 Halving error, 68, 80 Hauksson, H A., 192 Heath-Jarrow-Morton model, 6, 354 Heavy tails, heavy tailed, etc., 25–26, 45, 47–49, 72, 75–78, 99, 131–132, 149, 151, 153, 157–158, 160, 163, 165–166, 168–170, 182, 190–191, 209, 225, 228, 334, 341, 345, 353 Hendricks, D., 336, 340, 353 Heteroskedasticity, 134 Hill estimator, 159, 197–201 ‘Hill happy plot’, 199–200 ‘Hill horror plot’, 198–199 Hill plot, 198–200 Hinkley, D V., 107–108 Hisata, T., 316 Histogram, 10, 75–76, 83–84, 86–87, 89–90, 98, 111–113, 116, 241, 356 Historical kernel approach, 116 Historical scenarios, 298–299 Historical volatility, 127–130 Historical correlation, 138–139 Ho, T., 6, 265 Ho-Lee model, 228 Hodges, S., 43 Hoppe, R., 13 Hosking, J R M., 197 ‘Hot spots’, 276 ‘hsespdfperc’ function (in MMR Toolbox), 89 ‘hsvar’ function (in MMR Toolbox), 87 ‘hsvaresplot2D cl’ function (in MMR Toolbox), 88 ‘hsvarfigure’ function (in MMR Toolbox), 57, 85 HS (historical simulation) approaches, 10, 93, 96–98, 116–117, 274, 358 HS ES, 83–85, 89–92, 94 HS VaR, 56–58, 83–86, 89–92, 94, 103 Hua, P., 305 Hull, J., 94, 95, 162, 177–178, 181, 183, 216, 218, 224 Huschens, S., 164 Hydrology, 189 Hyperbolic distributions, 50, 167 Hysteria factor, 52 384 Index IGARCH, 131, 134–135 Ihle, G., 338 Illinski, K., 171 Illiquidity, 17, 20, 136, 301, 309, 323, 353 Implementation risk, 13, 31, 354–355 Implied correlation, 141 Implied covariance, 141 ‘Implied views’, 276–277 Implied volatility, 127, 136–137, 361 Importance sampling, 108, 219–220, 255, 263 Incremental coherent risk measures, 277 Incremental risks, 265–277 Incrementality, 271–273 Independence, iid, etc., 51, 92, 97, 110, 134, 148, 165, 174, 186, 190, 204–206, 221, 224, 241, 327, 329–331, 333–336, 338, 344 Independence (product) copula, 147, 181 Independent risk oversight, 362–363 Insurance portfolios, risks, etc., 15–16, 184, 227, 240–244 ‘insurancevar’ function (in MMR Toolbox), 241 ‘insurancevares’ function (in MMR Toolbox), 242 Integral, integration, etc., 80–81 Integrated risk management, 20, 241 Internal model approach, 52, 291, 354 Interquartile range, 115 Interocular trauma test, 75 Inui, K., 69, 72–73 Inverse distribution function, 221, 222, 333 Itô’s lemma, 161–162, 228 Itô process, 162 IVaR, 265–274, 276 IVaR approximations, 270–271 Jäckel, P., 144, 210, 213 Jackknife, 105, 107 Jackknife-after-bootstrap, 109 Jakobsen, S., 237, 258 James, J., 237 Jamshidian, F., 231–233, 238 Jarque-Bera test, 334–335, 345 Jarrow, R A., 315–316 Jaschke, S., 43 Joint stationarity, 138 JP Morgan, 9–10 Johnson, N L., 170 Johnson family, 170 Jones, M C., 170 Jorion, P., 12, 13 Ju, X., 13, 356 Jump-diffusion process, 168–169, 254 Jump process, 228–230 Karatzas, I., 43 Kashima Oil, Kato, T., 351 Kearns, P., 205 Kendall, M G., 69, 70, 102 Kennedy, W J., Jr., 123 Kernel, 83, 86–89, 111, 113–117 Key rate duration, Kijima, M., 69, 72–73 Knight, J., 12 Koenker, R., 172 Kolmogorov-Smirnov (KS) test, 339, 344–349 Komunjer, I., 336 Krakovsky, 316 Kreinin, A., 144 Krenn, G., 297 Kreyszig, E., 81, 216 Kruskal-Wallis test, 343 ‘ksdensity’ function (in MATLAB Statistics Toolbox), 116 ‘ks test stat’ function (MMR Toolbox), 346–347 Küchler, U., 43 Kuhn-Tucker conditions, 262 Kuiper test, 348–349 ‘kuiper test stat’ function (MMR Toolbox), 349 Kupiec, P., 301, 324, 339, 342 Kurtosis, 24–25, 46, 48–49, 62, 75–77, 109–110, 132, 151, 153, 159–160, 163, 167–172, 177, 186, 264, 323, 334–335, 343, 345 Kuruc, A., 12 L-estimator, 73, 104 Lagrangian, 262 Lakhany, A., 170 Landsman, Z., 154, 177 ‘Large’ command (in Excel), 56, 84 Latin hypercube, 222–223, 255 Lattice (binomial, etc.) approaches to options pricing, 215, 225, 254 Lattice approaches to options risk measurement, 225, 256 Laurent, S., 170 LaR (Liquidity-at-risk), 309, 316–319 Law invariance, 39 Lawrence, C., 10, 313 Least squares (LS), 115, 123, 124, 154, 158–159, 167 Le Saout, E., 313 Lee, B., 12 Lee, J., 299 Lee, Y.-S., 171 Leeson, N., 356 Levene test, 343 Leverage effect, 97 Lexicographic preferences, 32 Levin, A., 144 L´evy distribution, 47, 49–51, 72, 146, 165–167, 179, 191, 228 Lhabitant, 351 Index Life insurance risks, 242–244 Likelihood ratio, 195 check category Lilliefors test, 347–348 Limited liability, 156 Lin, C H., 170 Lin, K.-C., 249, 251 Lin, T.-K., 171 Linear homogeneity, 272–273 Linear programming, 37 Linsmeier, T J., 11 Liquidity, 16–17, 30, 234, 294, 300–301, 309–320, 353 Liquidity costs, 310, 312 Liquidity discount approach, 315–316 Liquidity risk(s), 11, 13, 17, 33, 286, 291–293, 299, 309–320 Liquidity spectrum, 16–17 Litterman, R., 12, 265, 276–277 Ljung-Box test, 132, 134 Lo, A., 313 Location parameter, 24, 47, 49–50, 160, 165, 167, 190, 177, 201, 205, 345 Log-t approach, 163 Lognormal distribution, 61, 62–64, 151, 158, 161–164, 184, 191, 202, 235–236, 252, 314, 351 Lognormal VaR, 61–64, 161–164 ‘lognormalvarfigure’ (in MMR Toolbox), 63, 163 ‘lognpdf’ function (in MATLAB Statistics Toolbox), 61 Longerstaey, J., 10 Longin, F., 189, 197 Longstaff, F A., 225 Lopez, J A., 143, 337–338, 342, 354 Lorentzian distribution, See under Cauchy distribution Low discrepancy methods, See under Quasi-random number methods Lower partial moment, 26, 39 LR test, etc., 329–330, 333–334 LTCM, 4, 299, 355 Lucas, A., 12, 354 Lüthi, H J., 263 LVaR (liquidity-adjusted VaR), 309–316, 319 MA process, 336 Macro hedging, 12 Malvergne, Y., 177, 181–182 Managing model risk, 358–363 Mandelbrot, B., 165 Manganelli, S., 172, 330 Mantegna, R N., 165 Mapping, 83, 88, 118, 263, 267–268, 279–290 Marginal distributions, 51, 145–147, 149, 178, 181 Marginal ES, 277 385 Marginal VaR, 268, 273 Mark (Deutschmark), 282 Mark to market, 15–17, 317, 354 Mark to model, 16–17, 354 ‘Market beta’ mapping, 283 Market portfolio, 8, 23 Market crash, 295, 298 Markov Chain Monte Carlo, 224, 230 Markowitz, H, M., Marshall, C., 13, 355 Martingale, 237–238 MATLAB, 22, 56, 116, 256, 325–328 Mausser, H., 104, 160 Maximum, maxima, etc., 42, 45, 76–77, 190, 193 Maximum copula, 147 Maximum domain of attraction, 201 Maximum likelihood, 115, 123–124, 132, 149, 154, 158–159, 165, 167–168, 177, 195–196, 202–204, 333 Maximum loss optimisation, 292, 306 McNeil, A J., 165, 189, 205, 207 Mean, 7, 20–22, 24, 26–27, 46, 48–49, 57–59, 61–68, 75–77, 89, 97, 105, 108–109, 139, 153–161, 163–164, 167–169, 171–173, 176, 178, 181, 184–186, 214, 223, 229, 235–236, 245, 252, 268, 282, 323, 343, 331–332, 334, 341, 357–358 Mean absolute deviation (MAD), 46 Mean excess function (MEF), 79, 203 Mean integrated square error, 114 Mean-reversion, 132, 214, 228–231 , 237 Mean square error (MSE), 114, 200 Mean-variance framework, 7, 19, 22–23, 25–27, 32, 44, 271 Mechanical stress tests, testing, etc., 293, 303–306 Median, 46, 89, 103, 106 Merton, R C., 168 Metallgesellschaft, 4, 301, 317 Mezrich, J., 135 Middle (risk) office, 362 Milevsky, M., 246 Mina, J., 258 Minimum, 45, 76–77, 190 Minimum copula, 147 Miranda, M S., 81 Mittnik, S., 166 MMR Toolbox, 41, 56, 65, 67–68, 87, 241, 256, 346–347, 349 Mode, 46–47, 76, 89, 343 ‘Model creep’, 356 Model risk, 17, 31, 110, 182, 187, 194, 212, 351–363 Model validation, vetting, etc., 321, 361–363 Modigliani-Miller theorem, 386 Index Moments, 24, 47–48, 108, 171, 177, 195–197, 223–224, 264 Moment-generating function, 47–48 Moment-matching, methods of moments, etc., 24, 81, 158, 195–197, 203, 223–224, 227–255 Monte Carlo simulation, 10, 74, 164–165, 169, 178, 181, 187, 209–249, 253–256, 263–264, 274, 296, 312, 336, 338, 340–341, 346–349, 354, 356, 358–359 Moosa, I A., 12, 137 Morgan Guaranty Trust Company, 130 Mori, A., 258 Mortality risks, 242–245 Multinomial distribution, 232 Multivariate distributions, 51, 145, 179, 206 Multivariate elliptical distributions, 145–146, 177–178, 183, 271 Multivariate EVT, extremes, etc., 148, 204, 206 Multivariate EWMA, 142 Multivariate GARCH, 97, 142–143 Multivariate generalised hyperbolic distribution, 177 Multivariate hyperbolic distribution, 177 Multivariate normal inverse Gaussian, 177 Multivariate normality, 122, 124, 145, 149, 151, 173–174, 176–178, 183, 269, 276, 304 Multivariate standard normality, 181 Multivariate t distribution, 146, 149, 176–177 Muranaga, J., 312 Naïve estimator, 86, 89, 111–113 Natural hedges, 265, 272 NatWest Bank, 4, 355 Near normal distributions, 8, 11, 19–20 Nelson, R B., 135, 147 Neural network approaches, 98–99, 221, 340 New York Bankers Association, New York Stock Exchange, Newton-Cotes methods, 80–81 Newton-Raphson methods, 137 Ng, V., 135 Niffikeer, C I., 237 Nijman, T., 186 Non-normality, 19–20, 27, 75, 99, 170–171 Non-parametric density estimation, 86–89, 111–117 Non-parametric statistics, 343, 356 Non-parametric methods, 74, 83–101, 104, 149, 184 Normal (Gaussian) copula, 148, 149 Normal (Gaussian) distribution, normality, etc., 8, 11, 19–22, 24–25, 46, 49–50, 57–61, 64, 70, 77, 100, 102–104, 106, 109, 115, 128, 132, 146, 151–154, 156–158, 160–171, 173, 178, 181, 184, 191, 195, 197, 202, 204, 207, 214, 228, 257, 260, 268, 282, 297, 301, 304, 312, 314–315, 331, 332, 334–335, 343–346, 348, 357–359 Normal ES, 64, 154, 155, 157 Normal inverse Gaussian (NIG) distribution, 50, 167 Normal mixture distributions, 50, 132, 167–168 Normal quantiles, 78–79 Normal VaR, 30, 49, 57–61, 64–65, 154–157, 159, 164, 170–171, 173, 209, 302 ‘normalesplot£D’ (in MMR Toolbox), 38 ‘normal spectral risk measure plot’ function (in MMR Toolbox), 41 ‘normalvar’ function (in MMR Toolbox), 65, 67–68 ‘normalvaresfigure’ function (in MMR Toolbox), 36, 65, 155 ‘normalvaresplot2D cl’ function (in MMR Toolbox), 37 ‘normalvarplot2D cl’ function (in MMR Toolbox), 30 ‘normalvarplot2D hp’ function (in MMR Toolbox), 156 ‘normalvarplot3D’ function (in MMR Toolbox), 31 ‘normalvarfigure’ (in MMR Toolbox), 28, 59 ‘norminv’ function (in MATLAB), 21 ‘normsinv’ function (in Excel), 21 Null hypothesis, 77, 332, 323, 325–327, 334–335, 341, 343–345 Numerical integration (or numerical quadrature), 41, 80, 81, 209 O’Brien, J., 353 October ‘87 crash, 3, 9, 298–300, 319, 353 Ohsawa, M., 313 Oliver, P., 179 Operational risk, 1, 11, 13, 15, 52 Options, 3, 15, 32, 125, 137, 141, 209, 210–211, 215, 218–220, 227, 236, 238, 249–264, 274, 290, 296, 298, 300, 302–305, 317–318, 351, 355–356 Orange County, 4, 13, 356 Order statistics, 73, 89, 91, 95, 98, 102–104, 164–165, 211, 349 Organised exchanges, 280, 291 Ornstein-Uhlenbeck process, 231 Orthogonal GARCH, 143 Outliers, 46, 75, 79, 322 Overlapping forecast periods, 335–336 Pagan, A R., 205 Pan, J., 95, 168–169 Paolella, M S., 166 Parametric density estimation, 111 Parameter-equality tests, 343–344 Parameter estimation, 158–159 Index Parameter risk, 110, 182, 187, 357–358 Parametric long-term VaR, 184–185 Parametric methods, 57, 74, 84, 99, 104, 149, 151–152, 154, 182, 184, 189, 274, 331 Pareto distribution, 191 Parmalat, 4, 299 ‘pcaes’ function (in MATLAB), 124 ‘pcaprelim’ function (in MATLAB), 122 ‘pcavar’ function (in MMR Toolbox), 123 Peaks over Threshold (POT) theory, 201–204 Pearson, N D., 11, 13, 356 Pearson family distributions, 170 Pellizon, L., 169 Pension portfolios, risks, etc., 16, 184, 227, 244–248 PensionMetrics, 244 Pension-ratio VaR, 248 percentile, 45, 89, 93, 102, 116, 171, 178, 192, 221, 260, 263, 332 ‘Percentile’ function (in Excel), 56 percentile interval approach, 107 Phelan, M J., 280 Pickands estimator, 197 Poisson process, 168, 229 Portfolio insurance, 353 Portfolio theory, 7–8, 10–11, 20, 24–25, 270 Portfolio-level analysis, 74, 151 Position limits, 12, 268, 274, 293, 306, 363 Position-level analysis, 74, 151, 178, 182 Positive definiteness, 118, 142, 143–144, 216 Positive semi-definiteness, 118, 142, 143–144, 216 Potters, M., 260–261 Power law tail, 165, 191 ‘prctile’ function (in MATLAB), 56 Precision (of estimators), 53, 69, 72, 212, 215 Predicted distribution, 343, 348 Preliminary data analysis, 77–79, 111, 321, 323 Present value, 5, 53, 238, 248, 289 Price elasticity of demand, 314 Principal components, PCA, etc., 118–125, 143–144, 222, 230–234, 238, 248, 281 Principal components ES, 121, 124 Principal components VaR, 121, 123 Prause, K., 177 Prinzler, R., 99 Pritsker, M., 92, 94, 98, 217, 263 Probability density functions (or pdf), 21, 24, 45–50, 70, 87, 114, 151, 158, 165, 167, 170, 191, 318 Probability functions, 45–51 Proctor and Gamble, Profile likelihood approach, 165 Programming problems, 355–356 Pseudo-random numbers, 81 PV01, 284–285 387 Put option, 253 Put-call parity, 362 QPS (quadratic probability score), 337–338 QQ plot, 75, 77–79, 134, 323 Quadratic programming, 260–263 Quantifying model risk, 21, 357–359 Quantile, 21, 27, 34–35, 38, 41, 43, 45, 68–72, 102–105, 116, 152, 160, 166, 192, 194–195, 201, 205, 335, 346 Quantile regression, 99, 172–173, 221 Quanto option, 141 Quasi-Bayesian maximum likelihood, 168 Quasi-random number methods, 81, 210 Rachev, S., 166 Random walk, 127, 237–238 RANDU generator, 213 Range, 323 Ranking models, 321, 336–338 Rauscher, F A., 99 Realised volatility, 137 Reference instruments, 279 Regime-switching Markov Chain, 169 Regression, 123, 195–196, 219, 334 Regulatory backtesting requirements, 34, 52, 328 Reinsurance treaties, 35 Reiss, R.-D., 102, 189 Resample, 85–86, 90, 91, 107, 109 Reserve requirements, 241 Reverse engineering, 281 Rho, 8, 258, 319 Richardson, M., 93 Richardson extrapolation, 68–69, 72, 104 Risk aggregation, 12, 20, 352 Risk audits, 362 Risk aversion, risk-averse, etc., 38–42, 66, 313, 348 Risk aversion function, 38–40, 66 Risk decomposition, 104, 265–277 Risk factors, 124, 144, 225, 272, 279–280, 290, 294–296, 299, 301, 303–305, 352 Risk-free asset, 22–23, 235–236 Risk-free return, 7, 306 Risk-loving, 39 Risk-neutrality, risk-neutral pricing, etc., 39, 210–211 Risk target, 12, 32 RiskMetrics, 9–11, 142, 280, 355 RiskMetrics Technical Document, 130, 280, 284 Robinson, C., 10, 313 Robust estimation methods, 158–159 Rockafellar, R T., 37 Rohatgi inequality, 264 Rosenblatt transformation, etc., 331–333, 335, 342 388 Index Rosenblatt-Berkowitz transformed data, 335–336, 344–345 Ross, S A., Rouvinez, C., 259–260, 263, 303–304 Rouvinez delta-gamma approach, 259–260 Rowe, D., 284 Runs tests, 330, 344 Safety-first criterion, 26 Saladin, T., 189 Sampling error, variation etc., 86, 108, 210, 228, 352 Sarbanes-Oxley Act, 362 Scaillet, O., 149 Scale invariance, 50, 147 Scale parameter, 24, 47, 49–50, 160, 165, 167, 177, 181, 190, 192, 201–202, 205, 345 Scenario analysis, 6, 292–293, 297–303, 320 Scenario catalogue, 299 Scenario simulation, 232–234, 238, 248 Scenarios as coherent risk measures, 42–43 Schachter, B., 87, 90, 116, 258, 291, 293, 306 Schaefer, S M., 260, 262–263, 304 Schwartz, E S., 225 ‘Seed’ number, 213 Self-similarity, 50, 165 Semi-parametric methods, 83–84, 98–100, 106, 127, 158–159, 184, 195, 197–200, 202 Sentana, E., 12 Shape parameter, 47, 181, 190, 202 Shapiro, S., 14 Shapiro-Francia test, 344 Shapiro-Wilks test, 344, 349 Sharpe, W., Shaw, J., 361 Sheedy, E., 284 Shimko, D., 92 Short-selling constraints, 23 Showa Shell, Siegel, M., 13, 355 Siegel-Tukey test, 343 Silverman, B W., 114–115 Simpson’s rule, 68, 81 Sin, C.-Y., 173 Singer, R., 318 Singular value decomposition, 216 Siu, T K., 357 Skew-t distribution, 170 Skewed distribution, 24, 26 Skewness, skew, etc., 24–25, 46, 48, 50, 61–62, 75–77, 99, 107, 111, 151, 153, 160, 165, 167–169, 171–172, 177, 334–335, 343–345 Sklar, A., 147 Solvency II, 52 Sornette, D., 177, 181–182 Soronow, D., 229 ‘Sort’ command (in MATLAB), 56 Sosa, J C., 237 SPAN (Standard Portfolio Analysis of Risk), 42, 291, 304 Spearman’s rank correlation, 146 Spectral risk measure, 37–42, 68 Spread option, 141 Square root of time rule, 50, 52, 127, 157, 184–187, 192 Stability, 50, 160, 165 Stability index (for L´evy distribution), 47, 49, 72 Standard deviation, 7, 10, 19–24, 26–27, 46–49, 57–59, 61–63, 65–68, 75–77, 105, 107, 139, 145, 153–161, 163–164, 171–172, 174–175, 181, 192, 214, 218, 221, 223, 235–237, 245, 252, 282, 294, 296, 304, 323, 331–332, 343, 357–358 Stahl, G., 144 Stand-alone VaR, 271, 265–276 Standard errors, 53, 69–73, 83, 107–108, 152, 224 Standard normal distribution, 21, 24–25, 49, 57, 71, 78–79, 85, 103, 107, 115, 121, 154, 160–161, 166–167, 174, 176, 178, 198–199, 213, 216, 221, 229, 263, 331, 333–334, 336, 344, 346 Stanley, H E., 165 Statistics Toolbox (in MATLAB), 61 Stevens, M A., 348 Stochastic dominance, 26, 36–37 Stochastic volatility, 153, 158, 163, 169, 214, 218, 353 Stop-loss strategy, 234–235 Stratified sampling, 220–224, 255 Stress tests, stress testing, etc., 17, 207, 261, 291–307, 311, 342, 359, 363 Stress VaR Stretched exponential distribution, 181 Strickland, C., 211, 218, 230 Stuart, A., 69–70, 102 Student t-distribution, 47, 76–77, 79, 159–160, 177, 343 Studer, G., 258, 263, 303–304 Stylised scenarios, 298 Subadditivity, 33–34, 37 Subjective (Bayesian) uncertainty, 224, 358 Subramanian, A., 315–316, Summary statistics, 75–77 Survival probabilities, 242–244 Swaps, 125, 236–237, 287, 290, 317–318 Swaptions, 236, 355 Symmetric distribution, 24–25, 46, 48, 76 Tail conditional expectation, 35 Tail conditional VaR, 35 Tail dependence, 149 Tail index, 159, 190, 192–195, 197, 201–203, 205, Index 389 Tail VaR, 35 Taleb, N., 13, 14 Tasche, D., 37, 265, 277 Taylor, J W., 99, 172 Taylor series expansion, 55, 267 — First order approximation, 257, 267–268, 270 — Second order approximation, 258, 259 t-copula, 148 t-distribution (see also under Student-t distribution), 25, 50, 102–103, 132, 146, 148, 151, 156, 158–160, 171–172, 177, 191, 202, 228, 311, 332, 358 t-tests, 343 t-VaR, 159–160, 171–172 Theta, 8, 258, 319 Thin tails, thin tailed, etc., 75, 77–78 Thomas, M., 189 Threshold GARCH, 135 Tibiletti, L., 265 Tibshirani, R J., 107–108 Tilman, L M., 340 Time to first exceedance test, 324 Time value of money, 53 Tippett, M., 170 Transactions costs, 313 ‘Transformation-into-Normality’ approach (HW), 177–178, 181, 183 Trapezoidal rule, 68, 80–81 Triangular kernel, 86, 114, 116 Truncated L´evy flight, 166 Tsay, R S., 123 Tunaru, R., 97 Turnbull, S M., 255 Type I error, 323, 341 Type II error, 323, 341 209, 211–212, 218, 220–221, 224, 234–235, 238–242, 246–247, 249–276, 281–284, 286–287, 291–295, 297, 301–304, 306–307, 311–318, 321–323, 325–328, 331, 336–339, 341, 351, 353–358 VaR-based decision rules, 12, 32, 36 VaR-beta, 272–273, 276 VaR bounds, 263–264 VaR decomposition, 265–277 VaR-defined remuneration, 32, 356 VaR disclosure (or reporting), 12, 29, 265 VaR distribution function, 89, 102–104 VaR elasticity, 273 Variable kernel, 115 Variance, 7, 20, 22, 26, 46–48, 50, 70, 105–109, 114, 119–120, 122, 124–125, 127, 132, 134, 138, 142, 146, 151, 154, 159–160, 165–169, 173, 200, 207, 211, 219, 274, 281–283, 287, 319, 334–335, 343, Variance-covariance matrix, See under Covariance matrix Variance-ratio test, 335, 343–344 Variance reduction methods, 108, 217–224, 238, 248, 255–256 Vega, 8, 249, 300, 319 Venkataraman, S., 167–168 Vlaar, P J G., 237 Volatility, 10, 64, 83, 91, 95–98, 100, 118, 124–125, 127–137, 140–143, 151, 156, 159, 169, 177, 184–187, 209, 213, 229, 237–238, 252, 279, 281–286, 294–295, 298, 302, 304, 306, 312, 315, 323, 352, 354–357 Volatility clustering, 91, 131, 151, 156, 163 Volatility smile, 137 Volatility term structure, 133–134 Volatility-weighted HS, 94–96, 98–99 Umantsev, L., 172 Unconditional distribution, 25–26, 45–46, 151–152, 204–205 Uniform distribution, 332–333, 336 Uniform random numbers, 220, 222 Uryasev, S., 37 US Annuity 2000 Basic Table (1996), 243 Utility function, 23, 32, 40, 43–44 Wakeman, L., 255, 320 Walter, C A., 143, 354 Wan, H., 224 Wang, C., 224 Wang, S S., 43 Wang, T., 43 Wang transform, 43 Weatherstone, D., Webber, N., 237 Wee, L.-S., 299 Weibull distribution, 191 Weighted HS, 92–98 White, A., 94, 95, 177, 178, 181, 183, 218, 224 Whitelaw, R., 93 Wiener, Z., 258 Wiener process, 161–162, 213, 228–229 Wilcoxon signed ranks test, 343 Wilmott, P., 305, 319 Vasicek model, 228 Valdez, E A., 154, 177 Valuation, 20 Van der Waerden test, 343 VaR (Value at Risk), 4, 9–14, 17, 19–20, 27–32, 34–40, 44–46, 49, 51–54, 56–64, 66–69, 71–74, 83–86, 88–95, 97–99, 100, 102–104, 108, 111–117, 119, 121, 123, 146, 151–152, 154–165, 167–168, 172–174, 176–177, 179–182, 184–185, 189, 193–194, 201–205, 390 Index Wilson, T C., 261–262, 304 Wilson (QP) delta-gamma approach, 261–263, 303 Wong, A C M., 258 Woods, M., 12 WorldCom, Worst-case scenario analysis, 43, 304 Worst conditional expectation, 35 Yakult Honsha, Yamai, Y., 26, 37, 72, 277, 316 Yoshiba, T., 26, 37, 72, 277, 351 Zangari, P., 10, 167–170, 258 Zero-arbitrage, 296, 303, 362 Zhu, Y., 231–233, 238 Zigrand, J.-P., 14 ... exchange rates) Market risks, in turn, can be classified into interest-rate risks, equity risks, exchange rate risks, commodity price risks, and so on, depending on whether the risk factor is... particular form of financial risk – namely, market risk, or the risk of loss (or gain) arising from unexpected changes in market prices (e.g., such as security prices) or market rates (e.g., such... Dynamic risks Fixed-income risks 9.4.1 Distinctive features of fixed-income problems 9.4.2 Estimating fixed-income risk measures Credit-related risks Insurance risks 9.6.1 General insurance risks